Estimating the spread of a market maker

Question

If we have an order book and we assume that we know there is only one market maker, how can we determine exactly the spread of the market maker? What if there are more than one market makers?

Answer 1

Model Building

I will answer this question from a statistical perspective since that has a definitive answer (under stated assumptions)

assumption 1) Suppose that each market-maker applies the same, unknown, spread, $X$ to quote a specific product.

assumption 2) Suppose that the uncertainty in the mid-market price of each marker is determined by a normal distribution about a mean with a variance, $\mathcal{N}(\mu, \sigma^2)$

Then the quoted bid-ask spread is determined by:

• Bid: Maximum mid - $\frac{X}{2}$
• Ask: Minimum mid + $\frac{X}{2}$

The expectation of this can be assessed and depends upon the number, $n$, of market makers. This answer: https://math.stackexchange.com/questions/473229/expected-value-for-maximum-of-n-normal-random-variable is helpful. We can then derive the result:

• $n = 1$: the tradeable bid-ask is $X$
• $n = 2$: the expected tradeable bid-ask is: $$\underbrace{\left ( \mu - \sigma \pi^{-\frac{1}{2}} + \frac{X}{2} \right )}_{ask} - \underbrace{\left ( \mu + \sigma \pi^{-\frac{1}{2}} - \frac{X}{2} \right )}_{bid} = X - 2\sigma\pi^{-\frac{1}{2}}$$
• $n = 3$: the expected tradeable bid-ask is: $X - 3\sigma\pi^{-\frac{1}{2}}$

The approximation given in the above link suggests that, in general, the expected tradeable bid-ask is given by:

$$ X - \sqrt{8 log(n)} \sigma$$

Reflection

I actually think this is quite interesting. Take for example a market I know well - interest rate swaps - where there might be say 10 market makers and the tradeable bid-ask is typically 0.1 bps. If the uncertainty standard deviation is considered the same value then using the above approximation that each market maker would have an individual bid-ask of 0.38bps, which might be a bit wide but not unreasonably.